Abstract: The search for extra dimensions is a challenging endeavor to probe physics
beyond the Standard Model. The joint detection of gravity waves (GW) and
electromagnetic (EM) signals from the merging of a binary system of compact
objects like neutron stars (NS), can help constrain the geometry of extra
dimensions. In particular, if our observable Universe is a 3+1 hypersurface, or
brane, embedded in a higher 4+1 Anti-de Sitter (AdS$_5$) spacetime, in which
gravity is the only field that propagates through the infinite bulk space,
while other fields are confined to the brane, then GW and EM signals between
two points on the brane would, in general, travel different paths. This would
result in a time-lag between the detection of GW and EM signals emitted
simultaneously from the same source. Assuming the standard $\Lambda$-Cold Dark
Matter ($\Lambda$CDM) scenario and performing a conservative analysis in which
we assume that the time-lag between the event GW170817 by the LIGO/VIRGO
collaborations and the GRB170817A signal detected by the Fermi Gamma-ray Burst
Monitor collaboration is entirely due to astrophysical processes associated
with the NS-NS merger, we set a bound on the AdS$_5$ radius of curvature $\ell
\lesssim 46\,$kpc at $68\%$ confidence level. Although the bound is not
competitive with current Solar System constraints, it is the first time that
data from a multi-messenger GW-EM measurement can be used to constrain the
physics of extra dimensions. Thus, our work provides a proof-of-principle of
the possibility of using multi-messenger astronomy for probing the geometry of
our space-time.